You are 8, which is a young age to learn about abstraction and generalization.

Exponents started out as a way to abbreviate expressions like 7 * 7 * 7 * 7 * 7. That takes time to write, and it is easy to forget one of the 7's and screw things up. So the exponent was born.

\(\displaystyle 7^5 = 7 * 7 * 7 * 7 * 7.\)

It started just as a way to make notation briefer and less prone to error. Make sense?

However that means

\(\displaystyle 7^3 * 7^4 = (7 * 7 * 7) * (7 * 7 * 7 * 7) = 7 * 7 * 7 * 7 * 7 * 7 * 7 = 7^7 = 7^{3+4}.\)

Then people made a discovery. It turns out that it doesn't make any difference whether the base number is 7 or 11 or 1/2, and it does not make any difference whether the exponents are 3 and 4 or 7 and 11. It is universally true that if a, b, and c are **ANY** numbers greater than zero and exponents mean what we have said they mean then

\(\displaystyle a^b * a^c = a^{b+c}.\)

If you want to test this **FIRST LAW OF EXPONENTS** on your calculator, go ahead.

\(\displaystyle 2^3 = 2 * 2 * 2 = 8.\\

2^5 =2 * 2 * 2 * 2 *2 = 4 * 4 * 2 = 16 * 2 = 32.\\

\text {So } 2^3 * 2^5 = 8 * 32 = 256. \implies 2^3 * 2^5 = 8 * 32 = 256.\\

2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 = 4 * 2 * 2 * 2 * 2 * 2 * 2 = 8 * 2 * 2 * 2 * 2 * 2 =\\

16 * 2 * 2 * 2 * 2 = 32 * 2 * 2 * 2 = 64 * 2 * 2 = 128 * 2 = 256.\)

So exponents are a handy way to save time and avoid errors by reducing work. Then someone figured out that

\(\displaystyle (2^3)^4 = (2 * 2 * 2) * (2 * 2 * 2) * (2 * 2 * 2) * (2 * 2 * 2) =\\

2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 = 2^{12} = 2^{3 * 4}.\)

Again it turns out that the principle is true no matter what the base or exponents are. We then get the

**SECOND LAW OF EXPONENTS**, namely, if a, b, and c are **ANY** numbers greater than zero,

\(\displaystyle (a^b)^c = a^{b*c}.\)

Did you follow all this?

If so, I should tell you that with a major adjustment to the definition of exponent, we get to keep the meanings you already know and the two laws shown above, but we get some additional neat ways to simplify our work and notation. So the modern mathematician does not define exponents the way you have learned. The modern definition gives the same results as your definition, but gives extra results as well.